F. Heartbeat
time limit per test
5 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

For an array $$$u_1, u_2, \ldots, u_n$$$, define

  • a prefix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j<i$$$;
  • a suffix maximum as an index $$$i$$$ such that $$$u_i>u_j$$$ for all $$$j>i$$$;
  • an ascent as an index $$$i$$$ ($$$i>1$$$) such that $$$u_i>u_{i-1}$$$.

You are given three cost arrays: $$$[a_1, a_2, \ldots, a_n]$$$, $$$[b_1, b_2, \ldots, b_n]$$$, and $$$[c_0, c_1, \ldots, c_{n-1}]$$$.

Define the cost of an array that has $$$x$$$ prefix maximums, $$$y$$$ suffix maximums, and $$$z$$$ ascents as $$$a_x\cdot b_y\cdot c_z$$$.

Let the sum of costs of all permutations of $$$1,2,\ldots,n$$$ be $$$f(n)$$$. Find $$$f(1)$$$, $$$f(2)$$$, ..., $$$f(n)$$$ modulo $$$998\,244\,353$$$.

Input

The first line contains an integer $$$n$$$ ($$$1\le n\le 700$$$).

The second line contains $$$n$$$ integers $$$a_1,\ldots,a_n$$$ ($$$0\le a_i<998\,244\,353$$$).

The third line contains $$$n$$$ integers $$$b_1,\ldots,b_n$$$ ($$$0\le b_i<998\,244\,353$$$).

The fourth line contains $$$n$$$ integers $$$c_0,\ldots,c_{n-1}$$$ ($$$0\le c_i<998\,244\,353$$$).

Output

Print $$$n$$$ integers: the $$$i$$$-th one is $$$f(i)$$$ modulo $$$998\,244\,353$$$.

Examples
Input
3
1 1 1
1 1 1
1 1 1
Output
1 2 6
Input
3
1 2 3
2 3 1
3 1 2
Output
6 13 34
Input
5
1 4 2 5 3
2 5 1 3 4
300000000 100000000 500000000 400000000 200000000
Output
600000000 303511294 612289529 324650937 947905622
Note

In the second example:

  • Consider permutation $$$[1,2,3]$$$. Indices $$$1,2,3$$$ are prefix maximums. Index $$$3$$$ is the only suffix maximum. Indices $$$2,3$$$ are ascents. In conclusion, it has $$$3$$$ prefix maximums, $$$1$$$ suffix maximums, and $$$2$$$ ascents. Therefore, its cost is $$$a_3b_1c_2=12$$$.
  • Permutation $$$[1,3,2]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
  • Permutation $$$[2,1,3]$$$ has $$$2$$$ prefix maximums, $$$1$$$ suffix maximum, and $$$1$$$ ascent. Its cost is $$$4$$$.
  • Permutation $$$[2,3,1]$$$ has $$$2$$$ prefix maximums, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$6$$$.
  • Permutation $$$[3,1,2]$$$ has $$$1$$$ prefix maximum, $$$2$$$ suffix maximums, and $$$1$$$ ascent. Its cost is $$$3$$$.
  • Permutation $$$[3,2,1]$$$ has $$$1$$$ prefix maximum, $$$3$$$ suffix maximums, and $$$0$$$ ascents. Its cost is $$$3$$$.

The sum of all permutations' costs is $$$34$$$, so $$$f(3)=34$$$.